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%% hw4.tex
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%% Started on  Tue Oct  4 14:19:52 2011 alex
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\section*{Homework 4}\renewcommand{\leftmark}{Homework 4}\phantomsection\addcontentsline{toc}{section}{Homework 4}
\begin{exercise}
Show that 
$$\displaystyle\prod^{\infty}_{n=2}\left(1-\frac{1}{n^{2}}\right)=\frac{1}{2}.$$
Relate this to the infinite product formula for sine.
\end{exercise}
\begin{exercise}
Show that $\displaystyle\prod^{\infty}_{n=2}\left(1-\frac{2}{n^{3}+1}\right)=\frac{2}{3}$.
\end{exercise}
\begin{exercise}
It is obvious that if $x$ is real (and not a pole of $\Gamma$),
then $\Gamma(x)$ is real. What is the set $\{x \in\RR \lst
\Gamma(x) > 0\}$? (The answer can be seen on the picture on page
415, but a proof is requested.) 
\end{exercise}
\begin{exercise}
Find explicitly
$\displaystyle\prod^{\infty}_{n=1}\left(1+\frac{1}{n^{2}}\right).$
\end{exercise}
